Arch
Calculate elliptical arch focus points, focal distance, semi-axes lengths, and approximate perimeter from span and rise. Free online arch calculator for construction, woodworking, and home renovation projects.
About This Calculator
The Elliptical Arch Calculator helps architects, carpenters, builders, and DIY enthusiasts determine the precise dimensions needed to construct and draw an elliptical arch. Whether you are framing a doorway, designing a window opening, or building an arched passageway, this calculator provides the key measurements required for accurate layout and construction.
The calculator uses the standard elliptical arch formula based on ellipse geometry. Given the span (base width, 2a) and rise (height, b), where a is the semi-major axis and b is the semi-minor axis, the focal distance is calculated as c = √(a² - b²). The focus points, located at distance c from the center on the baseline, are essential for the string method of drawing the elliptical curve. The approximate perimeter uses Ramanujan's formula: P ≈ π × (3(a + b) - √((3a + b)(a + 3b))). The rise must always be less than half the span to form a valid elliptical arch.
The string method for drawing an elliptical arch is a centuries-old technique still used today. First, mark the focus points on the base line at distance c from center. Place nails at the two focus points and at the top of the arch (rise). Loop a taught string around all three nails. Remove the top nail and, keeping the string tight, use a pencil to trace the elliptical curve from one side to the other. The total string length equals the full span.
Elliptical arches are preferred over semi-circular arches in many modern applications because they offer a lower profile while maintaining structural integrity. They distribute loads through compression along their curved shape, reducing horizontal thrust on supporting walls compared to segmental arches.
Regional Notes
India (IN): Dimensions are entered in centimeters or millimeters. Common residential arch spans range from 60-120 cm for interior doorways. Consult a structural engineer for load-bearing arch openings wider than 180 cm.
United States (US): Dimensions are entered in inches. Standard interior door openings are 32-36 inches wide. For drywall archways, use metal framing track bent to the elliptical curve with drywall applied over it.
United Kingdom (UK): Dimensions are entered in millimeters or centimeters. Standard UK door openings are typically 762-838 mm wide. For brick arches, consult a bricklayer for proper voussoir (wedge brick) placement.
Frequently Asked Questions
What is an elliptical arch?
An elliptical arch is an arch shaped as half of a horizontal ellipse. Unlike semi-circular arches that form a perfect half-circle, elliptical arches can be designed with a lower rise-to-span ratio, making them ideal for doorways, windows, and openings where ceiling clearance is limited.
How do I calculate the focus points of an elliptical arch?
Given the span (2a) and rise (b), the focal distance is c = √(a² - b²) where a is half the span. The focus points are located at distance c to the left and right from the center of the base. These points are used with the string method to draw the arch.
What is the string method for drawing an elliptical arch?
The string method involves placing nails at the two focus points and the top of the arch, looping a string around them, then removing the top nail and using the string as a guide to trace the ellipse. The total string length equals the span of the arch.
What is the difference between a semi-circular arch and an elliptical arch?
A semi-circular arch has a rise equal to half its span (forming a perfect half-circle), while an elliptical arch has a rise less than half the span. Elliptical arches are lower-profile and better suited for spaces with limited vertical clearance above doorways and windows.
What materials are used to build elliptical arches?
Elliptical arches can be built with drywall and metal framing for interior openings, brick or stone masonry for exterior features, or wood framing with plywood templates. Modern construction often uses pre-fabricated arched door frames or custom-built curved headers.
How do I frame an elliptical arch opening?
First, calculate the focus points using this calculator. Mark the span on your header, find the center, and mark the focus points. Install a plywood template cut to the elliptical curve, then frame around it with curved headers and studs following the template.
Can elliptical arches be used for load-bearing walls?
Yes, properly constructed elliptical arches distribute loads through compression along their curved profile. However, load-bearing arch construction typically requires engineering approval, proper keystone placement in masonry arches, and may need reinforced concrete or steel lintels for wider spans.
What is the typical rise-to-span ratio for an elliptical arch?
Elliptical arches typically have a rise-to-span ratio between 1:4 and 1:3. For example, a 48-inch span might have a 12-16 inch rise. The ratio cannot exceed 1:2, as that would be a semi-circular arch, and lower ratios produce flatter, more subtle curves.